What's the difference between "anti-virus", "anti-malware" and "anti-spyware" tools?

They're mostly synonymous as they are various forms of malicious software (malware). "Malware" is more broad whereas "virus" and "spyware" are more drilled down types of malware.

• Virus is defined as "the defining characteristic of viruses is that they are self-replicating computer programs which install themselves without the user's consent."
• Spyware is defined as "software that aids in gathering information about a person or organization without their knowledge and that may send such information to another entity without the consumer's consent, or that asserts control over a computer without the consumer's knowledge."
• Malware is defined as "short for malicious software, is software used to disrupt computer operation, gather sensitive information, or gain access to private computer systems."

Generally speaking, the main differences between the antivirus makers is that some actively listen and scan files as they are loaded into memory or visible in the folder you are viewing (MSE, Avast, etc.) whereas, AFAIK, others aren't active and work to rid you of the baddies when you execute them (Malware bytes, most "Spyware" programs).

If you are looking into purchasing some software, I would read reviews on how many resources they require and if you can limit that. I know that MSE is free and you can limit CPU use, in my experience it has been pretty solid. However, at work we have McAfee and my computer is essentially unusable while it is scanning, and domain policy limits my ability to limit it in traditional ways.

It may be a good question (if it isn't already) to ask "How does someone pick a good antivirus software" or something of that nature. If you are interested, please be sure to search first!

In terms of function, they're largely the same; they all have a database of known targets that they look for in the usual ways, e.g. monitoring file access, process memory, etc.

The difference is the type of thing that they're targetting. I can't tell you how exactly these terms are adhered to, but I can tell you the definitions:

• Malware: Any software that is malicious in nature. The term is mainly used as an umbrella term for viruses, trojans, adware, spyware, worms, or any other software that is likely to have a detrimental effect on the system.
• Virus: Technically speaking, viruses are types of malware that auto-replicate and infect new files on the system. However, the more general usage is to refer to explicitly malicious software, as opposed to just "potentially unwanted" or behaviour-collecting software.
• Spyware: Software that is used to collect information about you or your computer usage. This can be anything from information about what sites you visit and when, all the way to banking information and personal details.

Anti-malware and anti-virus software will almost certainly be very similar in terms of what they target. Anti-spyware software usually specifically targets spyware and adware - a good example is Spybot Free, which (at least currently) focuses entirely on spyware, and leaves the decision to install other anti-malware software (e.g. Windows Defender) to the user.